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Talk:Theta function
No sources again. Jiawhein \(a\)\(l\) 02:50, May 27, 2013 (UTC) :Scandalous! FB100Z • talk • 03:41, May 27, 2013 (UTC) ::What do you mean? Jiawhein \(a\)\(l\) 03:42, May 27, 2013 (UTC) :::Come on, sourcing isn't a serious business. We could add sources later. ☁ I want more ⛅ 04:50, May 27, 2013 (UTC) ::::☁ blue and ☁ red. And can show me the list of codes of the 2 texts? Jiawhein \(a\)\(l\) 08:49, May 27, 2013 (UTC) :::::the hell? FB100Z • talk • 17:37, May 27, 2013 (UTC) ::::::Nah, he (Jiawhein) was talking about my new signature. The cloud symbols were taken from the comments he posted on talk pages. Anyway, ☁ is U+2601, and ⛅ is U+26C5. -- ☁ I want more ⛅ 02:33, May 28, 2013 (UTC) This definition makes no sense. Is there a definition for \(\theta(0)\)? What is the "first fixed point after \(\alpha\)"? What's \(\theta(\alpha + \omega)\)? FB100Z • talk • 17:56, May 27, 2013 (UTC) :Assuming that every ordinal has "index", such as \(\epsilon_0\), \(\zeta_0\), \(\Gamma_0\), we can say that next fixed point-ordinal after \(\alpha_0\) is \(\alpha_{\alpha_{\alpha_{\alpha\cdots}}}\). Thus, \(\theta(\alpha+\omega)\) is \(\omega\) these fixed points after \(\theta(\alpha)\). I added it to the article. Ikosarakt1 (talk ^ ) 18:11, May 27, 2013 (UTC) ::Most indexings define normal functions, so \(\alpha_{\alpha_{\alpha_{\alpha\cdots}}}\) is fixed point then (fixed point theorem for normal functions; iteration converges to fixed point). LittlePeng9 (talk) 18:51, May 27, 2013 (UTC) This definition still makes little sense to me. Can someone point me to a paper formally defining the function? FB100Z • talk • 19:47, May 27, 2013 (UTC) :By the way, theta function is also commonly called Feferman's function or Feferman's theta function. I tried to find the paper, but found only wikipedian article. But we can note that Chris Bird made many comparisons between his array-separators and ordinals. Since his array notation is defined formally, we can try to define ordinals by analogies. Also, he defined theta function informally here. Ikosarakt1 (talk ^ ) 09:42, May 28, 2013 (UTC) Why not use Hyper-E notation to represent these power towers of omegas? It would make it more rigorous. For example, use E(\(\Omega\))n#\(\alpha\) Also, can someone please express ordinals such as \(\varepsilon_0\)using this? BTD6 maker (talk) 16:58, June 9, 2014 (UTC) :Well, you'd have to define how Hyper-E notation operates on ordinals (and it's surprisingly difficult; I've tried it). I don't see how replacing the theta function with something we haven't yet defined is more rigorous. :I do concede, however, that this article is terrible. There is a formal definition to the theta function, but for some reason we have a weird (and probably incorrect) homebrew on this page instead of the real deal you're.so. 17:15, June 9, 2014 (UTC) :: Hyper-E notation is just needed for two arguments. I thought that the ... to indicate a power tower is less rigorous as it didn't specify that the height remained the same. My definition is that: ::E(\(\Omega\))n#1 = \(\Omega\)^n ::E(\(\Omega\))n#(\(\alpha\)+1) = E(\(\Omega\))(E(\(\Omega\))n#\(\alpha\))#1 :: I do agree with you on your point about a general lack of formality. In rule 7 I found two identical rules regarding addition and an absence of one regarding multiplication. :While for me all definitions seem formally defined and logical. Multiplication can be removed at all without lose of the power as we can express \Omega*\omega^\alpha as \omega^{\Omega+\alpha} . It seems that me and FB100Z just have different concepts about formalization of recursive notations. For me a set of rules with string replacements is enough to call it formal. FB100Z thinks that anything have to be defined in set theory, but that looks non-intuitive and readers won't understand it. Ikosarakt1 (talk ^ ) 07:15, June 10, 2014 (UTC) ::I don't see why you want to remove multiplication. In my opinion, it's not natural to use exponentiation without multiplication. How would you even define exponentiation without it? LittlePeng9 (talk) 15:12, June 10, 2014 (UTC) :The rule \omega^{\alpha+1}n = \omega^\alpha+\omega^\alpha+\cdots+\omega^\alpha+\omega^\alpha with n \omega^\alpha 's works properly without multiplication. We can't use \Omega as the base then, but the power of the notation won't change. Ikosarakt1 (talk ^ ) 16:33, June 10, 2014 (UTC) ::But the question I want to know is: why are you all up and makin' your own rules? We should be using the definition employed by the function's original authors. you're.so. 19:05, June 10, 2014 (UTC)